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We prove that the asymptotics of ergodic integrals along an invariant foliation of a toral Anosov diffeomorphism, or of a pseudo-Anosov diffeomorphism on a compact orientable surface of higher genus, is determined (up to a logarithmic error) by the action of the diffeomorphism on the cohomology of the surface. As a consequence of our argument and of the results of Giulietti and Liverani [Parabolic dynamics and anisotropic Banach spaces.  J. Eur. Math. Soc. (JEMS)  21 (9) (2019), 2793–2858] on horospherical averages, toral Anosov diffeomorphisms have no Ruelle resonances in the open interval$(1, e^{h_{\mathrm {top}}})$   .

On the equidistribution of unstable curves for pseudo-Anosov diffeomorphisms of compact surfaces